Marginal Distribution Of Brownian Motion On Pseudo Sphere In Minkowski Space

The study of Brownian motion,named after British botanist Rorbert Brown who observed the random motion of particles suspended in water,was never stopped.The diffusion equation satisfied by the probability den-sity function of Brownian motion in Euclidean space was first introduced by physicists A.Einstein and Marian von Smoluchowski and was generalized to diffusion processes by Dutch scientist Adriaan Fokker and German physicist Max Planck.When studying the mathematical model of Brownian motion,the mathematicians found that the geometry of the space where a Browni-an particle stays impacts the recurrence of the diffusion process.After the proof of famous uniformization theorem by German mathematician P.Koebe and French mathematician H.Poincare,Brownian motion and Riemannian geometry were connected by potential theory,finally.With the help of special relativity constructed by A.Einstein,a lot of theories of classic thermostatistic were generalized to this new formulism.Un-der this structure of relativity,the probability density function p of Brownian motion in Minkowski space satisfies the diffusion equation while the marginal hitting distribution satisfies the stationary equation 1/2?gu = 0,where ?g is the Laplace-Beltrami operator in this Minkowski space.Thus,the calculation of marginal hitting distrabution of Brownian motion turns out to be solving a Dirichlet problem.In the first chapter,we introduce the above basic theories.Based on the above background,the goal of this paper is to discuss the marginal hit-ting distribution of Brownian motion on pseudo spheres in Minkowski space of different dimensions.Since the pseudo spheres in Minkowski space are hy-perboloid models,we attempt to apply the idea and method in[11],namely,the classic method of separation of variables.In the second chapter,the definitions of spacelike,timelike,lightlike do-mains and spacelike/timelike pseudo spheres in Minkowski space are stated,and the formulas of inner product,distance,metrics and Laplace-Beltrami operator in orthogonal coordinates are represented.The nonnegativity of spectrum of Laplace-Beltrami operator is also discussed.We calculate the expressions of Lapalce-Beltrami operators of 3 and higher dimensional cases in pseudo polar coordinates.Several special ordinary equations,hypergeometric function,Gegenbauer polynomials and special denotes are introduced at the end of this chapter.In the third chapter,3-dimensional Minkowski spaces are separated in two cases:R13 and R23.The Dirichlet problem is solved in detail with R13 and the formulas of the marginal hitting distributions are represented in Theorem 3.1.1 and Theorem 3.2.1.The probability properties of the hitting distribution are also discussed.The result of R23 is obtained directly in Theorem 3.3.1 because of the equivalence of the Dirichlet problems with the case of R13.Finally,in the fourth chapter,we discuss the cases of higher dimensional Minkowski spaces.According to the value of l and v,which leads to different degeneration of the Laplace operator on the unit sphere,the original Dirichlet problem is separated in 4 cases.Repeating the calculation in the 3-dimensional case,the marginal hitting distributions are obtained in Theorem 4.1.1 and Theorem 4.3.1 using method of separation of variables,boundary conditions and the orthogonality of trigonometric functions and Gegenbauer polynomials.The probability properties of every hitting distribution are discussed in detail for the spacelike cases.